Let $V$ be the vector space of continuous real-valued functions on $[0,1]$. Show that the function

$$\|f\|=\int_{0}^{1}|f(x)| d x$$

defines a norm on $V$.

For $n=1,2, \ldots$, let $f_{n}(x)=e^{-n x}$. Is $f_{n}$ a convergent sequence in the space $V$ with this norm? Justify your answer.